This is readily formalized (def. 3.1 below): If ##F## denotes the smooth manifold of “values” that the given kind of field may take at any spacetime point, then a field history ##\Phi## is modeled as a smooth function from spacetime to this space of values:

Given a section ##\Phi \in \Gamma_\Sigma(E)## of the field bundle (def. 3.1) and given a spacelike (def. 2.34) submanifold ##\Sigma_p \hookrightarrow \Sigma## (def. 3.34) of spacetime in codimension 1, then the restriction ##\Phi\vert_{\Sigma_p}## of ##\Phi## to ##\Sigma_p## may be thought of as a field configuration in space. As different spatial slices ##\Sigma_p## are chosen, one obtains such field configurations at different times. It is in this sense that the entirety of a section ##\Phi \in \Gamma_\Sigma(E)## is a history of field configurations, hence a field history (def 3.1).

After we obtain equations of motion below in def. 5.22, these serve as the “laws of nature” that field histories should obey, and they define the subspace of those field histories that do solve the equations of motion; this will be denoted

where the index ##\mu## ranges from ##0## to ##p##, while the index ##a## ranges from 1 to ##s##.

If this trivial vector bundle is regarded as a field bundle according to def. 3.1, then a field history ##\Phi## is equivalently an ##s##-tuple of real-valued smooth functions ##\Phi^a \colon \Sigma \to \mathbb{R}## on spacetime:

If ##id_{\mathbb{R}^n} \;\colon\; \mathbb{R}^n \to \mathbb{R}^n## is the identity function on ##\mathbb{R}^n##, then ##\left(id_{\mathbb{R}^n}\right)^\ast \;\colon\; X(\mathbb{R}^n) \to X(\mathbb{R}^n)## is the identity function on the set of plots ##X(\mathbb{R}^n)##;

If ##\mathbb{R}^{n_1} \overset{f}{\to} \mathbb{R}^{n_2} \overset{g}{\to} \mathbb{R}^{n_3}## are two composablesmooth functions between Cartesian spaces (def. 1.1), then pullback of plots along them consecutively equals the pullback along the composition:$$
f^\ast \circ g^\ast
=
(g \circ f)^\ast
$$

Once the step from smooth manifolds to diffeological spaces (def. 3.10) is made, characterizing the smooth structure of the space entirely by how we may probe it by mapping smooth Cartesian spaces into it, it becomes clear that the underlying set ##X_s## of a diffeological space ##X## is not actually crucial to support the concept: The space is already entirely defined structurally by the system of smooth plots it has, and the underlying set ##X_s## is recovered from these as the set of plots from the point ##\mathbb{R}^0##.

But we may just as well drop the mentioning of the underlying set ##X_s## in the definition of generalized smooth spaces. By simply stripping this requirement off of def. 3.10 we obtain the following more general and more useful definition (still “bosonic”, though, the supergeometric version is def. 3.40 below):

If ##id_{\mathbb{R}^n} \;\colon\; \mathbb{R}^n \to \mathbb{R}^n## is the identity function on ##\mathbb{R}^n##, then ##\left(id_{\mathbb{R}^n}\right)^\ast \;\colon\; X(\mathbb{R}^n) \to X(\mathbb{R}^n)## is the identity function on the set of plots ##X(\mathbb{R}^n)##.

If ##\mathbb{R}^{n_1} \overset{f}{\to} \mathbb{R}^{n_2} \overset{g}{\to} \mathbb{R}^{n_3}## are two composablesmooth functions between Cartesian spaces, then consecutive pullback of plots along them equals the pullback along the composition:$$
f^\ast \circ g^\ast
=
(g \circ f)^\ast
$$

Recall, for the next proposition 3.16, that in the definition 3.14 of a smooth set ##X## the sets ##X(\mathbb{R}^n)## are abstract sets which are to be thought of as would-be smooth functions “##\mathbb{R}^n \to X##”. Inside def. 3.14 this only makes sense in quotation marks, since inside that definition the smooth set ##X## is only being defined, so that inside that definition there is not yet an actual concept of smooth functions of the form “##\mathbb{R}^n \to X##”.

But now that the definition of smooth sets and of morphisms between them has been stated, and seeing that Cartesian space ##\mathbb{R}^n## are examples of smooth sets, by example 3.15, there is now an actual concept of smooth functions ##\mathbb{R}^n \to X##, namely as smooth sets. For the concept of smooth sets to be consistent, it ought to be true that this a posteriori concept of smooth functions from Cartesian spaces to smooth sets coincides wth the a priori concept, hence that we “may remove the quotation marks” in the above. The following proposition says that this is indeed the case:

from the set of homomorphisms of smooth sets from ##\mathbb{R}^n## (regarded as a smooth set via example 3.15) to ##X##, to the set of plots of ##X## over ##\mathbb{R}^n##, given by evaluating on the identity plot ##id_{\mathbb{R}^n}##.

for ##n \in \mathbb{N}## the set of plots from ##\mathbb{R}^n## to ##\mathbf{\Omega}^k## is the set of smooth differential k-forms on ##\mathbb{R}^n## (def. 1.18)$$
\mathbf{\Omega}^k(\mathbb{R}^n) \;:=\; \Omega^k(\mathbb{R}^n)
$$

For ##k = 1## the smooth set ##\mathbf{\Omega}^0## actually is a diffeological space, in fact under the identification of example 3.15 this is just the real line:

$$
\mathbf{\Omega}^0 \simeq \mathbb{R}^1
\,.
$$

But for ##k \geq 1## we have that the set of plots on ##\mathbb{R}^0 = \ast## is a singleton

$$
\mathbf{\Omega}^{k \geq 1}(\mathbb{R}^0) \simeq \{0\}
$$

consisting just of the zero differential form. The only diffeological space with this property is ##\mathbb{R}^0 = \ast## itself. But ##\mathbf{\Omega}^{k \geq 1}## is far from being that trivial: even though its would-be underlying set is a single point, for all ##n \geq k## it admits an infinite set of plots. Therefore the smooth sets ##\mathbf{\Omega}^k## for ##k \geq## are not diffeological spaces.

That the smooth set ##\mathbf{\Omega}^k## indeed deserves to be addressed as the universal moduli space of differential k-forms follows from prop. 3.16: The universal moduli space of ##k##-forms ought to carry a universal differential ##k##-forms ##\omega_{univ} \in \Omega^k(\mathbf{\Omega}^k)## such that every differential ##k##-form ##\omega## on any ##\mathbb{R}^n## arises as the pullback of differential forms of this universal one along some modulating morphism ##f_\omega \colon X \to \mathbf{\Omega}^k##:

for each smooth function ##f \;\colon\; \mathbb{R}^{n_1} \to \mathbb{R}^{n_2}## between Cartesian spaces the ordinary pullback of differential forms along ##f## is compatible with these choices, in that for every plot ##\mathbb{R}^{n_2} \overset{\Phi}{\to} X## we have$$
f^\ast\left(\Phi^\ast(\omega)\right)
=
( f^\ast \Phi )^\ast(\omega)
$$

is the differential form defined on any plot ##\mathbb{R}^n \overset{\Phi}{\to} X## as the ordinary exterior product of the pullback of th differential forms ##\omega_1## and ##\omega_2## to this plot:

But since there are of course more algebras ##A \in \mathbb{R}Algebras## than arise this way from smooth manifolds, we may turn this around and try to regard any algebra ##A## as defining a would-be space, which would have ##A## as its algebra of functions.

For example an infinitesimally thickened point should be a space which is “so small” that every smooth function ##f## on it which vanishes at the origin takes values so tiny that some finite power of them is not just even more tiny, but actually vanishes:

which is the tensor product of algebras of the algebra of smooth functions ##C^\infty(\mathbb{R}^n)## on an actual Cartesian space of some dimension ##n## (example 1.3), with an algebra of functions ##A \simeq_{\mathbb{R}} \langle 1\rangle \oplus V## of an infinitesimally thickened point, as above.

(This is sometimes called the algebra of dual numbers, for no good reason.) The underlying real vector space of this algebra is, as show, the direct sum of the multiples of 1 with the multiples of ##\epsilon##. A general element in this algebra is of the form

$$
a + b \epsilon \in (\mathbb{R}[\epsilon])/(\epsilon^2)
$$

where ##a,b \in \mathbb{R}## are real numbers. The product in this algebra is given by “multiplying out” as usual, and discarding all terms proportional to ##\epsilon^2##:

where ##a = f(0)## is the value of the function at the origin, and where ##b = \frac{\partial f}{\partial x}(0)## is its first derivative at the origin.

Therefore this algebra behaves like the algebra of smooth function on an infinitesimal neighbourhood ##\mathbb{D}^1## of ##0 \in \mathbb{R}## which is so tiny that its elements ##\epsilon \in \mathbb{D}^1 \hookrightarrow \mathbb{R}## become, upon squaring them, not just tinier, but actually zero:

The following example 3.22 shows that infinitesimal thickening is invisible for ordinary spaces when mapping out of these. In contrast example 3.23 further below shows that the morphisms into an ordinary space out of an infinitesimal space are interesting: these are tangent vectors and their higher order infinitesimal analogs.

Now this being an ##\mathbb{R}##-algebra homomorphism, its action on the multiples ##c \in \mathbb{R}## of the identity is fixed:

$$
f^\ast(1) = 1
\,.
$$

All the remaining elements are proportional to ##\epsilon##, and hence are nilpotent. However, by the homomorphism property of an algebra homomorphism it follows that it must send nilpotent elements ##\epsilon## to nilpotent elements ##f(\epsilon)##, because

which are the identity modulo ##\epsilon##. Such a morphism has to take any function ##f \in C^\infty(\mathbb{R}^n)## to

$$
f + (\partial f) \epsilon
$$

for some smooth function ##(\partial f) \in C^\infty(\mathbb{R}^n)##. The condition that this assignment makes an algebra homomorphism is equivalent to the statement that for all ##f_1,f_2 \in C^\infty(\mathbb{R}^n)## we have

We need to consider infinitesimally thickened spaces more general than the thickenings of just Cartesian spaces in def. 3.20. But just as Cartesian spaces (def. 1.1) serve as the local test geometries to induce the general concept of diffeological spaces and smooth sets (def. 3.14), so using infinitesimally thickened Cartesian spaces as test geometries immediately induces the corresponding generalization of smooth sets with infinitesimals:

of a morphism of infinitesimally thickened Cartesian spaces and of a plot of ##X##, as shown, subject to the equivalence relation which identifies two such pairs if there exists a smooth function ##f \colon \mathbb{R}^k \to \mathbb{R}^{k’}## such that

The “super”-terminology for something as down-to-earth as the mathematical principle behind the stability of matter may seem unfortunate. For better or worse, this terminology has become standard since the middle of the 20th century. But the concept that today is called supercommutative superalgebra was in fact first considered by Grassmann 1844 who got it right (“Ausdehnungslehre“) but apparently was too far ahead of his time and remained unappreciated.

For ##V## a finite dimensionalreal vector space, then the Grassmann algebra ##A := \wedge^\bullet_{\mathbb{R}} V^\ast## is a supercommutative superalgebra with ##A_{even} := \wedge^{even} V^\ast## and ##A_{odd} := \wedge^{odd} V^\ast##.More explicitly, if ##V = \mathbb{R}^s## is a Cartesian space with canonical dual coordinates ##(\theta^i)_{i = 1}^s## then the Grassmann algebra ##\wedge^\bullet (\mathbb{R}^s)^\ast## is the real algebra which is generated from the ##\theta^i## regarded in odd degree and hence subject to the relation

For ##A_1## and ##A_2## two supercommutative superalgebras, there is their tensor product supercommutative superalgebra ##A_1 \otimes_{\mathbb{R}} A_2##. For example for ##X## a smooth manifold with ordinary algebra of smooth functions ##C^\infty(X)## regarded as a supercommutative superalgebra by the first example above, the tensor product with a Grassmann algebra (second example above) is the supercommutative superalgebta$$
C^\infty(X) \otimes_{\mathbb{R}} \wedge^\bullet((\mathbb{R}^s)\ast)
$$

and extended from there as a bigraded derivation of bi-degree ##(1,even)##, in super-generalization of def. 1.19.

Accordingly, the operation of contraction with tangent vector fields (def. 1.20) has bi-degree ##(-1,\sigma)## if the tangent vector has super-degree ##\sigma##:

generator

bi-degree

##\phantom{AA} x^a##

(0,even)

##\phantom{AA} \theta^\alpha##

(0,odd)

##\phantom{AA} dx^a##

(1,even)

##\phantom{AA} d\theta^\alpha##

(1,odd)

derivation

bi-degree

##\phantom{AA} d##

(1,even)

##\phantom{AA}\iota_{\partial x^a}##

(-1, even)

##\phantom{AA}\iota_{\partial \theta^\alpha}##

(-1,odd)

This means that if ##\alpha \in \Omega^\bullet(\mathbb{R}^{b\vert s})## is in bidegree ##(n_\alpha, \sigma_\alpha)##, and ##\beta \in \Omega^\bullet(\mathbb{R}^{b \vert \sigma})## is in bidegree ##(n_\beta, \sigma_\beta)##, then

Hence there are two contributions to the sign picked up when exchanging two super-differential forms in the wedge product:

there is a “cohomological sign” which for commuting an ##n_1##-forms past an ##n_2##-form is ##(-1)^{n_1 n_2}##;

in addition there is a “super-grading” sign which for commuting a ##\sigma_1##-graded coordinate function past a ##\sigma_2##-graded coordinate function (possibly under the de Rham differential) is ##(-1)^{\sigma_1 \sigma_2}##.

from the set of homomorphisms of super smooth sets from ##\mathbb{R}^n \times \mathbb{D}## (regarded as a super smooth set via example 3.41) to ##X##, to the set of plots of ##X## over ##\mathbb{R}^n \times \mathbb{D}##, given by evaluating on the identity plot ##id_{\mathbb{R}^n \times \mathbb{D}}##.

Here in the third line we used that the Grassmann algebra ##\wedge^\bullet V^\ast## is free on its generators in ##V^\ast##, meaning that a homomorphism of supercommutative superalgebras out of the Grassmann algebra is uniquely fixed by the underlying degree-preserving linear function on these generators. Since in a Grassmann algebra all the generators are in odd degree, this is equivalently a linear map from ##V^\ast## to the odd-graded real vector space underlying ##C^\infty(U)[\theta](\theta^2)##, which is the direct sum ##C^\infty(U)_{odd} \oplus C^\infty(U)_{even}\langle \theta \rangle##.

That all these bijections are natural means that they are compatible with morphisms ##U \to U’## and therefore this says that ##[\mathbb{R}^{0\vert 1}, V_{odd}]## and ##V_{odd} \times V## are the same as seen by super-smooth plots, hence that they are isomorphic as super smooth sets.

Notice that these are ##\mathbb{K}##-valued odd functions: For instance if ##\mathbb{K} = \mathbb{C}## then each ##\chi_a## in turn has two components, a real part and an imaginary part.

A key point with the field bundle of the Dirac field (example 3.50) is that the field fiber coordinates ##(\psi^A)## or ##\left((\chi_a), (\xi^{\dagger \dot a})\right)## are now odd-graded elements in the function algebra on the field fiber, which is the Grassmann algebra ##C^\infty(S_{odd}) = \wedge^\bullet(S^\ast)##. Therefore they anti-commute with each other:

For ##U = \mathbb{R}^n## an ordinary Cartesian space (with no super-geometric thickening, def. 3.37) there is only a single ##U##-parameterized collection of field histories, hence a single plot$$
\Psi_{(-)}\;\colon\;\mathbb{R}^n \overset{ 0 }{\longrightarrow} \Gamma_\Sigma(\Sigma \times S_{odd})
$$

Moreover, these two kinds of plots determine the fermionic field space completely: It is in fact isomorphic, as a super vector space, to the bosonic field space shifted to odd degree (as in example 3.38):

Now, as in the first point above, here the first component is uniquely fixed to be the zero morphism ##\mathbb{R}^n \overset{0}{\to} S_{odd}##; and hence only the second component is free to choose. This is precisely the claim to be shown.

Proposition 3.51 how two basic facts about the Dirac field, which may superficially seem to be in tension with each other, are properly unified by supergeometry:

On the one hand a field history ##\Psi## of the Dirac field is not an ordinary section of an ordinary vector bundle. In particular its component functions ##\psi^A## anti-commute with each other, which is not the case for ordinary functions, and this is crucial for the Lagrangian density of the Dirac field to be well defined, we come to this below in example 5.8.

Therefore prop. 3.51 serves to shows how, even though a Dirac field is not defined to be an ordinary section of an ordinary vector bundle, it is nevertheless encoded by such an ordinary section: One says that this ordinary section is a “superfield-component” of the Dirac field, the one linear in a Grassmann variable ##\theta##.

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